Krasnoselskii-type algorithm for fixed points of multi-valued strictly pseudo-contractive mappings

Abstract

Let $q>1$ and let K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Let $T:K\to CB(K)$ be a multi-valued strictly pseudo-contractive map with a nonempty fixed point set. A Krasnoselskii-type iteration sequence $\{x_n\}$ is constructed and proved to be an approximate fixed point sequence of T, i.e., ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$. This result is then applied to prove strong convergence theorems for a fixed point of T under additional appropriate conditions. Our theorems improve several important well-known results.

MSC:47H04, 47H06, 47H15, 47H17, 47J25.

1 Introduction

For decades, the study of fixed point theory for multi-valued nonlinear mappings has attracted, and continues to attract, the interest of several well-known mathematicians (see, for example, Brouwer [1], Kakutani [2], Nash [3, 4], Geanakoplos [5], Nadla [6], Downing and Kirk [7]).

Interest in the study of fixed point theory for multi-valued maps stems, perhaps, mainly from the fact that many problems in some areas of mathematics such as in Game Theory and Market Economy and in Non-Smooth Differential Equations can be written as fixed point problems for multi-valued maps. We describe briefly the connection of fixed point theory for multi-valued mappings and these applications.

Game theory and market economy

In game theory and market economy, the existence of equilibrium was uniformly obtained by the application of a fixed point theorem. In fact, Nash [3, 4] showed the existence of equilibria for non-cooperative static games as a direct consequence of Brouwer [1] or Kakutani [2] fixed point theorem. More precisely, under some regularity conditions, given a game, there always exists a multi-valued map whose fixed points coincide with the equilibrium points of the game. A model example of such an application is the Nash equilibrium theorem (see, e.g., [3]).

Consider a game $G=(u_n,K_n)$ with N players denoted by n, $n=1,\dots ,N$, where $K_n\subset {\mathbb{R}}^{m_n}$ is the set of possible strategies of the n th player and is assumed to be nonempty, compact and convex and $u_n:K:=K_1\times K_2\cdots \times K_N\to \mathbb{R}$ is the payoff (or gain function) of the player n and is assumed to be continuous. The player n can take individual actions, represented by a vector ${\sigma }_n\in K_n$. All players together can take a collective action, which is a combined vector $\sigma =({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_N)$. For each n, $\sigma \in K$ and $z_n\in K_n$, we will use the following standard notations:

$$\begin{array}{c}{K}_{-n}:=K_1\times \cdots \times K_{n-1}\times K_{n+1}\times \cdots \times K_N,\hfill \\ {\sigma }_{-n}:=({\sigma }_1,\dots ,{\sigma }_{n-1},{\sigma }_{n+1},\dots ,{\sigma }_N),\hfill \\ (z_n,{\sigma }_{-n}):=({\sigma }_1,\dots ,{\sigma }_{n-1},z_n,{\sigma }_{n+1},\dots ,{\sigma }_N).\hfill \end{array}$$

A strategy ${\overline{\sigma }}_n\in K_n$ permits the n th player to maximize his gain under the condition that the remaining players have chosen their strategies ${\sigma }_{-n}$ if and only if

$$u_n({\overline{\sigma }}_n,{\sigma }_{-n})=\underset{z_n\in K_n}{max}{u}_n(z_n,{\sigma }_{-n}).$$

Now, let $T_n:K_{-n}\to 2^{K_n}$ be the multi-valued map defined by

$$T_n({\sigma }_{-n}):=\underset{z_n\in K_n}{Argmax}{u}_n(z_n,{\sigma }_{-n})\mathrm{\forall }{\sigma }_{-n}\in K_{-n}.$$

Definition A collective action $\overline{\sigma }=({\overline{\sigma }}_1,\dots ,{\overline{\sigma }}_N)\in K$ is called a Nash equilibrium point if, for each n, ${\overline{\sigma }}_n$ is the best response for the n th player to the action ${\overline{\sigma }}_{-n}$ made by the remaining players. That is, for each n,

$$u_n(\overline{\sigma })=\underset{z_n\in K_n}{max}{u}_n(z_n,{\overline{\sigma }}_{-n})$$

(1.1)

or, equivalently,

$${\overline{\sigma }}_n\in T_n({\overline{\sigma }}_{-n}).$$

(1.2)

This is equivalent to $\overline{\sigma }$ is a fixed point of the multi-valued map $T:K\to 2^K$ defined by

$$T(\sigma ):=[T_1({\sigma }_{-1}),T_2({\sigma }_{-2}),\dots ,T_N({\sigma }_{-N})].$$

From the point of view of social recognition, game theory is perhaps the most successful area of application of fixed point theory of multi-valued mappings. However, it has been remarked that the applications of this theory to equilibrium are mostly static: they enhance understanding of conditions under which equilibrium may be achieved but do not indicate how to construct a process starting from a non-equilibrium point and convergent to an equilibrium solution. This is part of the problem that is being addressed by iterative methods for a fixed point of multi-valued mappings.

Non-smooth differential equations

The mainstream of applications of fixed point theory for multi-valued maps has been initially motivated by the problem of differential equations (DEs) with discontinuous right-hand sides which gave birth to the existence theory of differential inclusion (DI). Here is a simple model for this type of application.

Consider the initial value problem

$$\frac{du}{dt}=f(t,u),\text{a.e. }t\in I:=[-a,a],u(0)=u_0.$$

(1.3)

If $f:I\times \mathbb{R}\to \mathbb{R}$ is discontinuous with bounded jumps, measurable in t, one looks for solutions in the sense of Filippov [8] which are solutions of the differential inclusion

$$\frac{du}{dt}\in F(t,u),\text{a.e. }t\in I,u(0)=u_0,$$

(1.4)

where

$$F(t,x)=[\underset{y\to x}{lim\hspace{0.17em}inf}f(t,y),\underset{y\to x}{lim\hspace{0.17em}sup}f(t,y)].$$

(1.5)

Now, set $H:=L^2(I)$ and let $N_F:H\to 2^H$ be the multi-valued Nemystkii operator defined by

$$N_F(u):=\{v\in H:v(t)\in F(t,u(t))\text{ a.e. }t\in I\}.$$

Finally, let $T:H\to 2^H$ be a multi-valued map defined by $T:=N_F\circ L^{-1}$, where $L^{-1}$ is the inverse of the derivative operator $Lu=u^{\prime }$ given by

$$L^{-1}v(t):=u_0+{\int }_0^{t}v(s)ds.$$

One can see that problem (1.4) reduces to the fixed point problem: $u\in Tu$.

Finally, a variety of fixed point theorems for multi-valued maps with nonempty and convex values is available to conclude the existence of a solution. We used a first-order differential equation as a model for simplicity of presentation, but this approach is most commonly used with respect to second-order boundary value problems for ordinary differential equations or partial differential equations. For more details about these topics, one can consult [9–12] and references therein as examples. Let E be a real normed linear space of dimension ≥2. The modulus of smoothness of E, ${\rho }_E$, is defined by

$${\rho }_E(\tau ):=sup\{\frac{\parallel x+y\parallel +\parallel x-y\parallel }{2}-1:\parallel x\parallel =1,\parallel y\parallel =\tau \},\tau >0.$$

A normed linear space E is called uniformly smooth if

$$\underset{\tau \to 0}{lim}\frac{{\rho }_E(\tau )}{\tau }=0.$$

It is well known (see, e.g., [13], p.16, [14]) that ${\rho }_E$ is nondecreasing. If there exist a constant $c>0$ and a real number $q>1$ such that ${\rho }_E(\tau )\le c{\tau }^q$, then E is said to be q-uniformly smooth. Typical examples of such spaces are the $L_p$, ${\ell }_p$ and $W_p^m$ spaces for $1<p<\mathrm{\infty }$, where

$$L_p(\text{or }{l}_p)\text{or}{W}_p^m\text{ is}\{\begin{array}{ll}2\text{-uniformly smooth}& \text{if }2\le p<\mathrm{\infty };\\ p\text{-uniformly smooth}& \text{if }1<p<2.\end{array}$$

Let $J_q$ denote the generalized duality mapping from E to $2^{E^{\ast }}$ defined by

$$J_q(x):=\{f\in E^{\ast }:〈x,f〉={\parallel x\parallel }^q\text{ and }\parallel f\parallel ={\parallel x\parallel }^{q-1}\},$$

where $〈\cdot ,\cdot 〉$ denotes the generalized duality pairing. $J_2$ is called the normalized duality mapping and is denoted by J. It is well known that if E is smooth, $J_q$ is single-valued.

Every uniformly smooth space has a uniformly Gâteaux differentiable norm (see, e.g., [13], p.17).

Let K be a nonempty subset of E. The set K is called proximinal (see, e.g., [15–17]) if for each $x\in E$, there exists $u\in K$ such that

$$d(x,u)=inf\{\parallel x-y\parallel :y\in K\}=d(x,K),$$

where $d(x,y)=\parallel x-y\parallel$ for all $x,y\in E$. Every nonempty, closed and convex subset of a real Hilbert space is proximinal. Let $CB(K)$ and $P(K)$ denote the families of nonempty, closed and bounded subsets and nonempty, proximinal and bounded subsets of K, respectively. The Hausdorff metric on $CB(K)$ is defined by

$$D(A,B)=max\{\underset{a\in A}{sup}d(a,B),\underset{b\in B}{sup}d(b,A)\}$$

for all $A,B\in CB(K)$. Let $T:D(T)\subseteq E\to CB(E)$ be a multi-valued mapping on E. A point $x\in D(T)$ is called a fixed point of T if $x\in Tx$. The fixed point set of T is denoted by $F(T):=\{x\in D(T):x\in Tx\}$.

A multi-valued mapping $T:D(T)\subseteq E\to CB(E)$ is called L-Lipschitzian if there exists $L>0$ such that

$$D(Tx,Ty)\le L\parallel x-y\parallel \mathrm{\forall }x,y\in D(T).$$

(1.6)

When $L\in (0,1)$ in (1.6), we say that T is a contraction, and T is called nonexpansive if $L=1$.

Definition 1.1 Let K be a nonempty subset of a real Hilbert space H. A map $T:K\to H$ is called k-strictly pseudo-contractive if there exists $k\in (0,1)$ such that

$${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+k{\parallel x-y-(Tx-Ty)\parallel }^2\mathrm{\forall }x,y\in K.$$

(1.7)

Browder and Petryshyn [18] introduced and studied the class of strictly pseudo-contractive maps as an important generalization of the class of nonexpansive maps (mappings $T:K\to K$ satisfying $\parallel Tx-Ty\parallel \le \parallel x-y\parallel \mathrm{\forall }x,y\in K$). It is trivial to see that every nonexpansive map is strictly pseudo-contractive.

Motivated by this, Chidume et al.[19] introduced the class of multi-valued strictly pseudo-contractive maps defined on a real Hilbert space H as follows.

Definition 1.2 A multi-valued map $T:D(T)\subset H\to CB(H)$ is called k-strictly pseudo-contractive if there exists $k\in (0,1)$ such that for all $x,y\in D(T)$,

$${(D(Tx,Ty))}^2\le {\parallel x-y\parallel }^2+k{\parallel x-y-(u-v)\parallel }^2\mathrm{\forall }u\in Tx,v\in Ty.$$

(1.8)

They then proved convergence theorems for approximating fixed points of multi-valued strictly pseudo-contractive maps (see [19]) which extend recent results from the class of multi-valued nonexpansive maps to the more general and important class of multi-valued strictly pseudo-contractive maps.

Single-valued strictly pseudo-contractive maps have also been defined and studied in real Banach spaces, which are much more general than Hilbert spaces.

Definition 1.3 Let K be a nonempty subset of a real normed space E. A map $T:K\to E$ is called k-strictly pseudo-contractive (see, e.g., [13], p.145, [18]) if there exists $k\in (0,1)$ such that for all $x,y\in K$, there exists $j(x-y)\in J(x-y)$ such that

$$〈Tx-Ty,j(x-y)〉\le {\parallel x-y\parallel }^2-k{\parallel x-y-(Tx-Ty)\parallel }^2.$$

(1.9)

In this paper, we define multi-valued strictly pseudo-contractive maps in arbitrary normed space E as follows.

Definition 1.4 A multi-valued map $T:D(T)\subset E\to CB(E)$ is called k-strictly pseudo-contractive if there exists $k\in (0,1)$ such that for all $x,y\in D(T)$,

$$k{(D(Ax,Ay))}^2\le 〈u-v,j(x-y)〉\mathrm{\forall }u\in Ax,v\in Ay,$$

(1.10)

where $A:=I-T$ and I is the identity map on E.

We observe that if T is single-valued, then inequality (1.10) reduces to (1.9).

Several papers deal with the problem of approximating fixed points of multi-valued nonexpansive mappings defined on Hilbert spaces (see, for example, Sastry and Babu [15], Panyanak [16], Song and Wong [17], Khan et al.[20], Abbas et al.[21] and the references contained therein) and their generalizations (see, e.g., Chidume et al.[19] and the references contained therein).

Chidume et al.[19] proved the following theorem for multi-valued k-strictly pseudo-contractive mappings defined on real Hilbert spaces.

Theorem CCDM (Theorem 3.2 [19])

Let K be a nonempty, closed and convex subset of a real Hilbert space H. Suppose that$T:K\to CB(K)$is a multi-valued k-strictly pseudo-contractive mapping such that$F(T)\ne \mathrm{\varnothing }$. Assume that$Tp=\{p\}$for all$p\in F(T)$. Let$\{x_n\}$be a sequence defined iteratively from$x_0\in K$by

$$x_{n+1}=(1-\lambda )x_n+\lambda y_n,$$

(1.11)

where$y_n\in T{x}_n$and$\lambda \in (0,1-k)$. Then${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$.

Using Theorem CCDM, Chidume et al. proved several convergence theorems for the approximation of fixed points of strictly pseudo-contractive maps under various additional mild compactness-type conditions either on the operator T or on the domain of T. The theorems proved in [19] are significant generalizations of several important results on Hilbert spaces (see, e.g., [19]).

Our purpose in this paper is to extend Theorem CCDM and other related results in [19], using Definition 1.4, from Hilbert spaces to the much more general class of q-uniformly smooth real Banach spaces. As we have noted, theses spaces include the $L_p$, $l_p$ and $W^{m,p}$ spaces, $1<p<\mathrm{\infty }$ and $m\ge 1$. Finally, we give important examples of multi-valued maps satisfying the conditions of our theorems.

2 Preliminaries

In the sequel, we need the following definitions and results.

Definition 2.1 Let E be a real Banach space and T be a multi-valued mapping. The multi-valued map $(I-T)$ is said to be strongly demiclosed at 0 (see, e.g., [22]) if for any sequence $\{x_n\}\subseteq D(T)$ such that $\{x_n\}$ converges strongly to $x^{\ast }$ and $d(x_n,T{x}_n)$ converges to 0, then $d(x^{\ast },T{x}^{\ast })=0$.

Lemma 2.2[19]

Let E be a reflexive real Banach space and let$A,B\in CB(X)$. Assume that B is weakly closed. Then, for every$a\in A$, there exists$b\in B$such that

$$\parallel a-b\parallel \le D(A,B).$$

(2.1)

Proposition 2.3 Let K be a nonempty subset of a real Banach space E and let$T:K\to CB(K)$be a multi-valued k-strictly pseudo-contractive mapping. Assume that for every$x\in K$, Tx is weakly closed. Then T is Lipschitzian.

Proof We first observe that for any $x\in D(T)$, the set Tx is weakly closed if and only if the set Ax is weakly closed. Now, let $x,y\in D(T)$ and $u\in Ax$. From Lemma 2.2, there exists $v\in Ay$ such that

$$\parallel u-v\parallel \le D(Ax,Ay).$$

(2.2)

Using the fact that T is k-strictly pseudo-contractive and inequality (2.2), we have

$$\begin{array}{rcl}k{(D(Ax,Ay))}^2& \le & 〈u-v,j(x-y)〉\\ \le & \parallel u-v\parallel \parallel x-y\parallel \\ \le & D(Ax,Ay)\parallel x-y\parallel .\end{array}$$

So,

$$D(Ax,Ay)\le \frac{1}{k}\parallel x-y\parallel \mathrm{\forall }x,y\in D(T).$$

(2.3)

From the definition of the Hausdorff distance, we have

$$D(Tx,Ty)\le D(Ax,Ay)+\parallel x-y\parallel \mathrm{\forall }x,y\in D(T).$$

(2.4)

Using (2.3) and (2.4), we obtain

$$D(Tx,Ty)\le L_k\parallel x-y\parallel \mathrm{\forall }x,y\in D(T),\text{ where }{L}_k:=\frac{1+k}{k}.$$

Therefore, T is $L_k$-Lipschitzian. □

Remark 1 We note that for a single-valued map T, for each $x\in D(T)$, the set Tx is always weakly closed.

Lemma 2.4 Let$q>1$, E be a q-uniformly smooth real Banach space, $k\in (0,1)$. Suppose$T:D(T)\subset E\to CB(E)$is a multi-valued map with$F(T)\ne \mathrm{\varnothing }$, and for all$x\in D(T)$, $x^{\ast }\in F(T)$,

$$k{\left(D(Ax,A{x}^{\ast })\right)}^2\le 〈u-v^{\ast },j(x-x^{\ast })〉\mathrm{\forall }u\in Ax,v^{\ast }\in A{x}^{\ast },$$

(2.5)

where$A:=I-T$, I is the identity map on E. If$T{x}^{\ast }=\{x^{\ast }\}$for all$x^{\ast }\in F(T)$, then the following inequality holds:

$$〈x-y,j_q(x-x^{\ast })〉\ge k^{q-1}{\parallel x-y\parallel }^q,\mathrm{\forall }x\in D(T),\mathrm{\forall }y\in Tx.$$

Proof Let $x\in D(T)$, $u\in Ax$, $x^{\ast }\in F(T)$. Then, from inequality (2.5), the definition of the Hausdorff metric and the assumption that $T{x}^{\ast }=\{x^{\ast }\}$, we have

$$k{\left(D(Ax,A{x}^{\ast })\right)}^2\le \parallel u\parallel \parallel x-x^{\ast }\parallel \le D(Ax,A{x}^{\ast })\parallel x-x^{\ast }\parallel .$$

So,

$$kD(Ax,A{x}^{\ast })\le \parallel x-x^{\ast }\parallel \mathrm{\forall }x\in D(T),x^{\ast }\in F(T).$$

(2.6)

Therefore, for all $x\in D(T)$, $y\in Tx$, $x^{\ast }\in F(T)$ such that $x\ne x^{\ast }$, using inequalities (2.5) and (2.6) and the fact that $j_q(x-x^{\ast })={\parallel x-x^{\ast }\parallel }^{q-2}j(x-x^{\ast })$, we obtain

$$\begin{array}{rcl}〈x-y,j_q(x-x^{\ast })〉& =& {\parallel x-x^{\ast }\parallel }^{q-2}〈x-y,j(x-x^{\ast })〉\\ \ge & k^{q-1}{\left(D(Ax,A{x}^{\ast })\right)}^q\\ \ge & k^{q-1}{\parallel x-y\parallel }^q.\end{array}$$

$D(Ax,A{x}^{\ast })\ge \parallel x-x^{\ast }\parallel$ since $T{x}^{\ast }=\{x^{\ast }\}$. This completes the proof. □

Lemma 2.5 Let K be a nonempty closed subset of a real Banach space E and let$T:K\to P(K)$be a k-strictly pseudo-contractive mapping. Assume that for every$x\in K$, Tx is weakly closed. Then$(I-T)$is strongly demiclosed at zero.

Proof Let $\{x_n\}\subseteq K$ be such that $x_n\to x$ and $d(x_n,T{x}_n)\to 0$ as $n\to \mathrm{\infty }$. Since K is closed, we have that $x\in K$. Since, for every n, $T{x}_n$ is proximinal, let $y_n\in T{x}_n$ such that $\parallel x_n-y_n\parallel =d(x_n,T{x}_n)$. Using Lemma 2.2, for each n, there exists $z_n\in Tx$ such that

$$\parallel y_n-z_n\parallel \le D(T{x}_n,Tx).$$

We then have

$$\begin{array}{rcl}\parallel x-z_n\parallel & \le & \parallel x-x_n\parallel +\parallel x_n-y_n\parallel +\parallel y_n-z_n\parallel \\ \le & \parallel x-x_n\parallel +\parallel x_n-y_n\parallel +D(T{x}_n,Tx)\\ \le & \parallel x-x_n\parallel +\parallel x_n-y_n\parallel +L_k\parallel x_n-x\parallel .\end{array}$$

Observing that $d(x,Tx)\le \parallel x-z_n\parallel$, it then follows that

$$d(x,Tx)\le \parallel x-x_n\parallel +\parallel x_n-y_n\parallel +L_k\parallel x_n-x\parallel .$$

Taking limit as $n\to \mathrm{\infty }$, we have that $d(x,Tx)=0$. Therefore $x\in Tx$. The proof is completed. □

Lemma 2.6[23]

Let$q>1$and E be a smooth real Banach space. Then the following are equivalent:

• (i) E is q-uniformly smooth.

• (ii) There exists a constant $d_q>0$ such that for all $x,y\in E$,

  $${\parallel x+y\parallel }^q\le {\parallel x\parallel }^q+q〈y,j_q(x)〉+d_q{\parallel y\parallel }^q.$$

• (iii) There exists a constant $c_q>0$ such that for all $x,y\in E$ and $\lambda \in [0,1]$,

  $${\parallel (1-\lambda )x+\lambda y\parallel }^q\ge (1-\lambda ){\parallel x\parallel }^q+\lambda {\parallel y\parallel }^q-w_q(\lambda )c_q{\parallel x-y\parallel }^q,$$

where$w_q(\lambda ):={\lambda }^q(1-\lambda )+\lambda {(1-\lambda )}^q$.

From now on, $d_q$ denotes the constant that appeared in Lemma 2.6. Let $\mu :=min\{1,{(\frac{q{k}^{q-1}}{d_q})}^{\frac{1}{q-1}}\}$.

3 Main results

We prove the following theorem.

Theorem 3.1 Let$q>1$be a real number and K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Suppose that$T:K\to CB(K)$is a multi-valued k-strictly pseudo-contractive mapping such that$F(T)\ne \mathrm{\varnothing }$and such that$Tp=\{p\}$for all$p\in F(T)$. For arbitrary$x_1\in K$and$\lambda \in (0,\mu )$, let$\{x_n\}$be a sequence defined iteratively by

$$x_{n+1}=(1-\lambda )x_n+\lambda y_n,$$

(3.1)

where$y_n\in T{x}_n$. Then${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$.

Proof Let $x^{\ast }\in F(T)$. Then, using the recursion formula (3.1), Lemmas 2.6 and 2.4, we have

$$\begin{array}{rcl}{\parallel x_{n+1}-x^{\ast }\parallel }^q& =& {\parallel x_n-x^{\ast }-\lambda (x_n-y_n)\parallel }^q\\ \le & {\parallel x_n-x^{\ast }\parallel }^q-\lambda q〈x_n-y_n,j_q(x_n-x^{\ast })〉+{\lambda }^q{d}_q{\parallel x_n-y_n\parallel }^q\\ \le & {\parallel x_n-x^{\ast }\parallel }^q-q\lambda k^{q-1}{\parallel x_n-y_n\parallel }^q+{\lambda }^q{d}_q{\parallel x_n-y_n\parallel }^q\\ =& {\parallel x_n-x^{\ast }\parallel }^q-\lambda (q{k}^{q-1}-d_q{\lambda }^{q-1}){\parallel x_n-y_n\parallel }^q.\end{array}$$

(3.2)

It follows that

$$\sum _{n=1}^{\mathrm{\infty }}{\parallel x_n-y_n\parallel }^q<\mathrm{\infty }.$$

Hence, ${lim}_{n\to \mathrm{\infty }}\parallel x_n-y_n\parallel =0$. Since $y_n\in T{x}_n$, we have that ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$. □

A mapping $T:K\to CB(K)$ is called hemicompact if, for any sequence $\{x_n\}$ in K such that $d(x_n,T{x}_n)\to 0$ as $n\to \mathrm{\infty }$, there exists a subsequence $\{x_{n_j}\}$ of $\{x_n\}$ such that $x_{n_j}\to p\in K$. We note that if K is compact, then every multi-valued mapping $T:K\to CB(K)$ is hemicompact.

We now prove the following corollaries of Theorem 3.1.

Corollary 3.2 Let$q>1$be a real number and K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Let$T:K\to CB(K)$be a multi-valued k-strictly pseudo-contractive mapping with$F(T)\ne \mathrm{\varnothing }$and such that$Tp=\{p\}$for all$p\in F(T)$. Suppose that T is continuous and hemicompact. Let$\{x_n\}$be a sequence defined iteratively from$x_1\in K$by

$$x_{n+1}=(1-\lambda )x_n+\lambda y_n,$$

(3.3)

where$y_n\in T{x}_n$and$\lambda \in (0,\mu )$. Then the sequence$\{x_n\}$converges strongly to a fixed point of T.

Proof From Theorem 3.1, we have ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$. Since T is hemicompact, there exists a subsequence $\{x_{n_j}\}$ of $\{x_n\}$ such that $x_{n_j}\to p$ for some $p\in K$. Since T is continuous, we have $d(x_{n_j},T{x}_{n_j})\to d(p,Tp)$. Therefore, $d(p,Tp)=0$ and so $p\in F(T)$. Setting $x^{\ast }=p$ in the proof of Theorem 3.1, it follows from inequality (3.2) that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exists. So, $\{x_n\}$ converges strongly to p. This completes the proof. □

Corollary 3.3 Let$q>1$be a real number and K be a nonempty, compact and convex subset of a q-uniformly smooth real Banach space E. Let$T:K\to CB(K)$be a multi-valued k-strictly pseudo-contractive mapping with$F(T)\ne \mathrm{\varnothing }$and such that$Tp=\{p\}$for all$p\in F(T)$. Suppose that T is continuous. Let$\{x_n\}$be a sequence defined iteratively from$x_1\in K$by

$$x_{n+1}=(1-\lambda )x_n+\lambda y_n,$$

(3.4)

where$y_n\in T{x}_n$and$\lambda \in (0,\mu )$. Then the sequence$\{x_n\}$converges strongly to a fixed point of T.

Proof Observing that if K is compact, every map $T:K\to CB(K)$ is hemicompact, the proof follows from Corollary 3.2. □

Remark 2 In Corollary 3.2, the continuity assumption on T can be dispensed if we assume that for every $x\in K$, the set Tx is proximinal and weakly closed. In fact, we have the following result.

Corollary 3.4 Let$q>1$be a real number and K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Let$T:K\to CB(K)$be a multi-valued k-strictly pseudo-contractive mapping with$F(T)\ne \mathrm{\varnothing }$and such that for every$x\in K$, Tx is weakly closed and$Tp=\{p\}$for all$p\in F(T)$. Suppose that T is hemicompact. Let$\{x_n\}$be a sequence defined iteratively from$x_1\in K$by

$$x_{n+1}=(1-\lambda )x_n+\lambda y_n,$$

(3.5)

where$y_n\in T{x}_n$and$\lambda \in (0,\mu )$. Then the sequence$\{x_n\}$converges strongly to a fixed point of T.

Proof Following the same arguments as in the proof of Corollary 3.2, we have $x_{n_j}\to p$ and ${lim}_{n\to \mathrm{\infty }}d(x_{n_j},T{x}_{n_j})=0$. Furthermore, from Lemma 2.5, $(I-T)$ is strongly demiclosed at zero. It then follows that $p\in Tp$. Setting $x^{\ast }=p$ and following the same computations as in the proof of Theorem 3.1, we have from inequality (3.2) that $lim\parallel x_n-p\parallel$ exists. Since $\{x_{n_j}\}$ converges strongly to p, it follows that $\{x_n\}$ converges strongly to $p\in F(T)$. The proof is completed. □

A mapping $T:K\to CB(K)$ is said to satisfy Condition (I) if there exists a strictly increasing function $f:[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ with $f(0)=0$, $f(r)>0$ for all $r\in (0,\mathrm{\infty })$ such that

$$d(x,T(x))\ge f\left(d(x,F(T))\right)\mathrm{\forall }x\in D.$$

Convergence theorems have been proved in real Hilbert spaces for multi-valued nonexpansive mappings T under the assumption that T satisfies Condition (I) (see, e.g., [16, 24]). The following corollary extends such theorems to multi-valued strictly pseudo-contractive maps and to q-uniformly smooth real Banach spaces.

Corollary 3.5 Let$q>1$be a real number and K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Let$T:K\to P(K)$be a multi-valued k-strictly pseudo-contractive mapping with$F(T)\ne \mathrm{\varnothing }$and such that for every$x\in K$, Tx is weakly closed and$Tp=\{p\}$for all$p\in F(T)$. Suppose that T satisfies Condition (I). Let$\{x_n\}$be a sequence defined iteratively from$x_1\in K$by

$$x_{n+1}=(1-\lambda )x_n+\lambda y_n,$$

(3.6)

where$y_n\in T{x}_n$and$\lambda \in (0,\mu )$. Then the sequence$\{x_n\}$converges strongly to a fixed point of T.

Proof From Theorem 3.1, we have ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$. Using the fact that T satisfies Condition (I), it follows that ${lim}_{n\to \mathrm{\infty }}f(d(x_n,F(T)))=0$. Thus there exist a subsequence $\{x_{n_j}\}$ of $\{x_n\}$ and a sequence $\{p_j\}\subset F(T)$ such that

$$\parallel x_{n_j}-p_j\parallel <\frac{1}{2^j}\mathrm{\forall }j\in \mathbb{N}.$$

By setting $x^{\ast }=p_j$ and following the same arguments as in the proof of Theorem 3.1, we obtain from inequality (3.2) that

$$\parallel x_{n_{j+1}}-p_j\parallel \le \parallel x_{n_j}-p_j\parallel <\frac{1}{2^j}.$$

We now show that $\{p_j\}$ is a Cauchy sequence in K. Notice that

$$\begin{array}{rcl}\parallel p_{j+1}-p_j\parallel & \le & \parallel p_{j+1}-x_{n_{j+1}}\parallel +\parallel x_{n_{j+1}}-p_j\parallel \\ <& \frac{1}{2^{j+1}}+\frac{1}{2^j}<\frac{1}{2^{j-1}}.\end{array}$$

This shows that $\{p_j\}$ is a Cauchy sequence in K and thus converges strongly to some $p\in K$. Using the fact that T is L-Lipschitzian and $p_j\to p$, we have

$$\begin{array}{rcl}d(p_j,Tp)& \le & D(T{p}_j,Tp)\\ \le & L\parallel p_j-p\parallel ,\end{array}$$

so that $d(p,Tp)=0$ and thus $p\in Tp$. Therefore, $p\in F(T)$ and $\{x_{n_j}\}$ converges strongly to p. Setting $x^{\ast }=p$ in the proof of Theorem 3.1, it follows from inequality (3.2) that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exists. So, $\{x_n\}$ converges strongly to p. This completes the proof. □

Corollary 3.6 Let$q>1$be a real number and K be a nonempty, compact and convex subset of a q uniformly smooth real Banach space E. Let$T:K\to P(K)$be a multi-valued k-strictly pseudo-contractive mapping with$F(T)\ne \mathrm{\varnothing }$and such that for every$x\in K$, the set Tx is weakly closed and$Tp=\{p\}$for all$p\in F(T)$. Let$\{x_n\}$be a sequence defined iteratively from$x_1\in K$by

$$x_{n+1}=(1-\lambda )x_n+\lambda y_n,$$

(3.7)

where$y_n\in T{x}_n$and$\lambda \in (0,\mu )$. Then the sequence$\{x_n\}$converges strongly to a fixed point of T.

Proof From Theorem 3.1, we have ${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$. Since $\{x_n\}\subseteq K$ and K is compact, $\{x_n\}$ has a subsequence $\{x_{n_j}\}$ that converges strongly to some $p\in K$. Furthermore, from Lemma 2.5, $(I-T)$ is strongly demiclosed at zero. It then follows that $p\in Tp$. Setting $x^{\ast }=p$ and following the same arguments as in the proof of Theorem 3.1, we have from inequality (3.2) that $lim\parallel x_n-p\parallel$ exists. Since $\{x_{n_j}\}$ converges strongly to q, it follows that $\{x_n\}$ converges strongly to $p\in F(T)$. This completes the proof. □

Corollary 3.7 Let$q>1$be a real number and K be a nonempty compact convex subset of a q uniformly smooth real Banach space E. Let$T:K\to P(K)$be a multi-valued nonexpansive mapping. Assume that$Tp=\{p\}$for all$p\in F(T)$. Let$\{x_n\}$be a sequence defined iteratively from$x_1\in K$,

$$x_{n+1}=(1-\lambda )x_n+\lambda y_n,$$

(3.8)

where$y_n\in T{x}_n$and$\lambda \in (0,\mu )$. Then the sequence$\{x_n\}$converges strongly to a fixed point of T.

Remark 3 The recursion formula (3.1) of Theorem 3.1 is of the Krasnoselkii type (see, e.g., [25]) and is known to be superior to the recursion formula of the Mann algorithm (see, e.g., Mann [26]) in the following sense: (i) The recursion formula (3.1) requires less computation time than the formula of the Mann algorithm because the parameter λ in formula (3.1) is fixed in $(0,1-k)$, whereas in the algorithm of Mann, λ is replaced by a sequence $\{c_n\}$ in $(0,1)$ satisfying the following conditions: ${\sum }_{n=1}^{\mathrm{\infty }}{c}_n=\mathrm{\infty }$, $lim{c}_n=0$. The $c_n$ must be computed at each step of the iteration process. (ii) The Krasnoselskii-type algorithm usually yields rate of convergence as fast as that of a geometric progression, whereas the Mann algorithm usually has order of convergence of the form $o(1/n)$.

Remark 4 In [24], the authors replace the condition $Tp=\{p\}$$\mathrm{\forall }p\in F(T)$ with the following restriction on the sequence $y_n:y_n\in P_T{x}_n$, i.e., $y_n\in T{x}_n$ and $\parallel y_n-x_n\parallel =d(x_n,T{x}_n)$. We observe that if, for example, the set $T{x}_n$ is a closed and convex subset of a real Hilbert space, then $y_n$ is unique and is characterized by

$$〈x_n-y_n,y_n-u_n〉\ge 0\mathrm{\forall }{u}_n\in T{x}_n.$$

Since this $y_n$ has to be computed at each step of the iteration process, this makes the recursion formula difficult to use in any possible application.

Remark 5 The addition of bounded error terms to the recursion formula (3.1) leads to no generalization.

Remark 6 Our theorems in this paper are important generalizations of several important recent results in the following sense: (i) Our theorems extend results proved for multi-valued nonexpansive mappings in real Hilbert spaces (see, e.g., [15–17, 20, 21]) to a much larger class of multi-valued strictly pseudo-contractive mappings and in a much larger class of q-uniformly smooth real Banach spaces. (ii) Our theorems are proved with the superior Krasnoselskii-type algorithm.

We give examples of multi-valued maps where, for each $x\in K$, the set Tx is proximinal and weakly closed.

Example 1 Let $f:\mathbb{R}\to \mathbb{R}$ be an increasing function. Define $T:\mathbb{R}\to 2^{\mathbb{R}}$ by

$$Tx=[f(x-),f(x+)]\mathrm{\forall }x\in \mathbb{R},$$

where $f(x-):={lim}_{y\to x^{-}}f(y)$ and $f(x+):={lim}_{y\to x^{+}}f(y)$. For every $x\in \mathbb{R}$, Tx is either a singleton or a closed and bounded interval. Therefore, Tx is always weakly closed and convex. Hence, for every $x\in \mathbb{R}$, the set Tx is proximinal and weakly closed.

Example 2 Let H be a real Hilbert space and $f:H\to \mathbb{R}$ be a convex continuous function. Let $T:H\to 2^H$ be the multi-valued map defined by

$$Tx=\partial f(x)\mathrm{\forall }x\in H,$$

where $\partial f(x)$ is the subdifferential of f at x and is defined by

$$\partial f(x)=\{z\in H:〈z,y-x〉\le f(y)-f(x)\mathrm{\forall }y\in H\}.$$

It is well known that for every $x\in H$, $\partial f(x)$ is nonempty, weakly closed and convex. Therefore, since H is a real Hilbert space, it then follows that for every $x\in H$, the set Tx is proximinal and weakly closed. The subdifferential has deep connection with convex optimization problems.

The condition $Tp=\{p\}$ for all $p\in F(T)$, which is imposed in all our theorems of this paper, can actually be replaced by another condition (see, e.g., Shahzad and Zegeye [24]). This is done in Theorem 3.9.

Let K be a nonempty, closed and convex subset of a real Hilbert space, $T:K\to P(K)$ be a multi-valued map and $P_T:K\to CB(K)$ be defined by

$$P_T(x):=\{y\in Tx:\parallel x-y\parallel =d(x,Tx)\}.$$

We will need the following result.

Lemma 3.8 (Song and Cho [27])

Let K be a nonempty subset of a real Banach space and$T:K\to P(K)$be a multi-valued map. Then the following are equivalent:

• (i) $x^{\ast }\in F(T)$;

• (ii) $P_T(x^{\ast })=\{x^{\ast }\}$;

• (iii) $x^{\ast }\in F(P_T)$. Moreover, $F(T)=F(P_T)$.

Remark 7 We observe from Lemma 3.8 that if $T:K\to P(K)$ is any multi-valued map with $F(T)\ne \mathrm{\varnothing }$, then the corresponding multi-valued map $P_T$ satisfies $P_T(p)=\{p\}$ for all $p\in F(P_T)$, the condition imposed in all our theorems and corollaries. Consequently, the examples of multi-valued maps $T:K\to CB(K)$ satisfying the condition $Tp=\{p\}$ for all $p\in F(T)$ abound.

Theorem 3.9 Let$q>1$be a real number and K be a nonempty, closed and convex subset of a q-uniformly smooth real Banach space E. Suppose that$T:K\to CB(K)$is a multi-valued mapping such that$F(T)\ne \mathrm{\varnothing }$. Assume that$P_T$is k-strictly pseudo-contractive. For arbitrary$x_1\in K$and$\lambda \in (0,\mu )$, let$\{x_n\}$be a sequence defined iteratively by

$$x_{n+1}=(1-\lambda )x_n+\lambda y_n,$$

where$y_n\in P_T(x_n)$. Then${lim}_{n\to \mathrm{\infty }}d(x_n,T{x}_n)=0$.

We conclude this paper with an example of a multi-valued map T for which $P_T$ is k-strictly pseudo-contractive, the condition assumed in Theorem 3.9. Trivially, every nonexpansive map is strictly pseudo-contractive.

Example 3 Let $E=\mathbb{R}$ with the usual metric and $T:\mathbb{R}\to CB(\mathbb{R})$ be the multi-valued map defined by

$$Tx=\{\begin{array}{ll}[0,\frac{x}{2}],& x\in (0,\mathrm{\infty }),\\ [\frac{x}{2},0],& x\in (-\mathrm{\infty },0].\end{array}$$

Then $P_T$ is strictly pseudo-contractive. In fact, $P_T(x)=\{\frac{x}{2}\}$ for all $x\in \mathbb{R}$.